广义Artin-Schreier多项式
Generalized Artin-Schreier polynomials
摘要：
令$F$是包含$F_{p^n}$作为子字段的素特征$p$的字段。我们将$q(X)=X^{p^n}-X-a\in F[X]$称为广义Artin-Schreier多项式。假设$q(X)$是不可约的，设$C_{q(X)}$是$q(X)$的伴随矩阵。然后$ad\, C_{q(X)}$有这样的极不寻常的性质，任何$A\in{\mathfrak{ gl}}(m)$，使$ad\, A$有类似的性质被证明是类似的同伴矩阵的不可约广义Artin-Schreier多项式。讨论了特征$p$域上1维李代数的不可分解模的张量积分解问题、与不可约广义Artin-Schreier多项式相联系的Galois扩张的每个中间域的显式本原元的求取问题、以及寻找多项式族不可约的充要条件的问题。
英文摘要：
Let $F$ be a field of prime characteristic $p$ containing $F_{p^n}$ as a subfield. We refer to $q(X)=X^{p^n}-X-a\in F[X]$ as a generalized Artin-Schreier polynomial. Suppose that $q(X)$ is irreducible and let $C_{q(X)}$ be the companion matrix of $q(X)$. Then $ad\, C_{q(X)}$ has such highly unusual properties that any $A\in{\mathfrak{ gl}}(m)$ such that $ad\, A$ has like properties is shown to be similar to the companion matrix of an irreducible generalized Artin-Schreier polynomial. We discuss close connections with the decomposition problem of the tensor product of indecomposable modules for a 1-dimensional Lie algebra over a field of characteristic $p$, the problem of finding an explicit primitive element for every intermediate field of the Galois extension associated to an irreducible generalized Artin-Schreier polynomial, and the problem of finding necessary and sufficient conditions for the irreducibility of a family of polynomials.
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